Topics in low-dimensional topology

长期计划的工作范围广泛的项目，探讨了一系列的几何，拓扑和数论问题。 大多数的问题要探讨的启发低维拓扑，虽然几个问题适合到一个更广泛的背景。 其中包括谱理论的性质和覆盖空间的同调性，复积算子的结构，迹场的算法，真实的射影结构的变形和极限，以及相似区间交换映射在几何无限曲面群中的作用。统一的主题是捆绑在一起的许多不同方面的双曲流形在低维和高维。 虽然佩雷尔曼的工作已经把拓扑学和几何学结合在一起，但我们对这种几何学的理解还有很长的路要走。 在这个项目中提出的方向上的进展将大大增加这种理解。一个空间被称为三维流形，如果它是由小块组成的，所有这些小块都是“像”我们生活在其中的普通三维空间。数学家知道如何解释“像”非常精确，有两个非常不同的概念，这是非常重要的，在这种情况下：“拓扑”和“几何”。 佩雷尔曼最近的工作验证了七十年代瑟斯顿的工作中出现的一个长期存在的猜想，即拓扑图和几何图密切相关。 这是全球理解的一个重要部分。 在几何流形中，最重要的是所谓的双曲流形。 这个类在数学的许多领域都是普遍存在的，从低维拓扑到动力系统，再到数论。 事实上，这门课在物理学中也至关重要。 然而，即使与Perlemann的工作，双曲流形本身仍然是相当不了解，虽然他们的重要性，使他们成为磁铁的研究远远超过三十年。 这一建议本身对双曲流形的结构研究方面和许多问题与他们有关。例如，他打算继续研究一个著名的古老猜想，即哪些二维物体可以存在于这些三维物体中。

Long plans to work on a wide spectrum of projects which explore a range of geometrical, topological and number-theoretical problems. Most of the issues to be explored are inspired by low dimensional topology, although several of the problems fit into a much wider context. These include the behavior of the spectral theory and homology of covering spaces, the structure of commensurators, the arithmetic of trace fields, the deformations and limits of real projective structures, and the role of similarity interval exchange maps in geometrically infinite surface groups. The unifying theme is the tying together of the many disparate aspects of hyperbolic manifolds in both low and higher dimensions. For while Perelman's work has drawn together the topological and geometric, our understanding of this geometry still has a long way to go. Progress in the directions proposed in this project would add significantly to this understanding.A space is called a 3-manifold if it is made of small chunks all of which are ``like'' the ordinary 3-dimensional space that we live in. Mathematicians understand how to interpret "like" very precisely and there are two very different notions which are of great importance in this setting: "topologically-like" and "geometrically-like". The recent work of Perelmann has verified a long-standing conjecture which emerged in the seventies with the work of Thurston, namely that the topological and geometrical pictures are intimately related. This is an important piece of global understanding. Amongst geometrical manifolds, by far the most important are the class called hyperbolic manifolds. This class is ubiquitous in many areas of mathematics, ranging from low-dimensional topology, to dynamical systems, to number theory. Indeed, this class is also crucially important in physics. However, even with Perlemann's work, hyperbolic manifolds themselves are still fairly poorly understood, although their importance have made them a magnet for research for well over thirty years. This proposal directs itself towards aspects of the structural study of hyperbolic manifolds and many issues related to them. For example, he intends to continue work on a famous old conjecture about which sorts of two-dimensional objects can live inside these 3-dimensional objects.